text: In case of an incompressible flow — for instance of a liquid, or a gas at low Mach numbers; but not for sound waves — the velocity  has zero divergence:
  with the dot denoting the inner product. As a result, the velocity potential  has to satisfy Laplace's equation
  where  is the Laplace operator (sometimes also written ). In this case the flow can be determined completely from its kinematics: the assumptions of irrotationality and zero divergence of flow. Dynamics only have to be applied afterwards, if one is interested in computing pressures: for instance for flow around airfoils through the use of Bernoulli's principle.
  In incompressible flows, contrary to common misconception, the potential flow indeed the satisfies the full Navier–Stokes equations, not just the Euler equations, because the viscous term 
  is identically zero. It is the inability of the potential flow to satisfy the required boundary conditions, especially near solid boundaries, makes it invalid in representing the required flow field. If the potential flow satisfies the necessary conditions, then it is the required solution of the incompressible Navier–Stokes equations.
  In two dimensions, potential flow reduces to a very simple system that is analyzed using complex analysis (see below).